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| '''Mahāvīra''' (or '''Mahaviracharya''', "Mahavira the Teacher") was a 9th-century [[Jain]] [[mathematician]] from [[Mysore]], [[India]].{{sfn|Pingree|1970}}{{sfn|O'Connor|Robertson|2000}}{{sfn|Tabak|2009|p=42}} He was the author of ''Gaṇitasārasan̄graha'' (or ''Ganita Sara Samgraha'', c. 850), which revised the [[Brāhmasphuṭasiddhānta]].{{sfn|Pingree|1970}} He was patronised by the [[Rashtrakuta]] king [[Amoghavarsha]].{{sfn|Puttaswamy|2012|p=231}} He separated astrology from mathematics. It is the earliest Indian text entirely devoted to mathematics.<ref>The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the ... by Clifford A. Pickover: page 88</ref> He expounded on the same subjects on which Aryabhata and Brahmagupta contended, but he expressed them more clearly. There are several points worth noting about the work of Mahavira. His work is a highly syncopated approach to algebra and the emphasis in much of his text is on developing the techniques necessary to solve algebraic problems.<ref>Algebra: Sets, Symbols, and the Language of Thought by John Tabak: p.43</ref> He is highly respected among Indian mathematicians, because of his establishment of terminology for concepts such as equilateral, and isosceles triangle; rhombus; circle and semicircle.<ref>Geometry in Ancient and Medieval India by T. A. Sarasvati Amma: page 122</ref> Mahāvīra's eminence spread in all South India and his books proved inspirational to other mathematicians in [[Southern India]].{{sfn|Hayashi|2013}} It was translated into [[Telugu language]] by [[Pavuluri Mallana]] as ''Saar Sangraha Ganitam''.<ref>Census of the Exact Sciences in Sanskrit by David Pingree: page 388</ref>
| | Just about 1 out of every 100 people suffers from this skin condition. People lives in country like USA, Australia and Canada known as the world. The abundance of genetic diseases associated with vitiligo clearly proves that there are a number of genes which normally govern the development and wellbeing of the melanocytes. It works by decreasing the excretion of melanin from the melanin producing cells in your skin. Even some renowned skin care and cosmetic products manufactures also claim to contain herbal oil and natural substances in their remedy. <br><br>In this connection, we will only discuss about the skin disorder when the immune system becomes dysfunctional. The signs of the white patches consist of depression or mood swings and of course the depigmentation of the vitiligo. Avoid cosmetics like cream, powder and other triggering agents. Vitiligo cure can be seen in many vitiligo skin patients who use the [http://discover-prague.info/sitemap/ vitiligo natural treatment]. Several herbal remedies employ various herbal components which contain psoralens, a group of naturally occurring substances. <br><br>Nevertheless, it is important to begin treatment early because older patches are harder to repigment. In US alone there are about 3 to 5 million vitiligo patients are suffering from this skin disorder disease. Infections: Infections may stimulate the onset of psoriasis. He also emphasized that he was more than proud to be black and that all the rumors that people spread were so untrue and did more than just hurt him. Re-pigmentation with Stop vitiligo Oil is very fast, good results are mostly noticeable after 2 to 3 weeks (May bee after some weeks) of common program. <br><br>But for nearly everyone people who have Vitiligo, depigmentation happens on different organs of the body (generalized pattern). Psoralea corylifolia (Babchi) is the main ingredient in the pigmento. Yellow spots, liver spots; urticaria worse from warmth of bed. He invented the Young Volumizer which is a safe procedure that is less invasive in reversing aging signs to reveal a natural and younger looking face. Your dysfunction might be owing to various tense results in. <br><br>Both conditions have a higher risk percentage in African Americans and both are immunological conditions. In most of the cases, a blend of transmissible, immunologic and ecological factors can trigger Vitiligo. Anti Vitiligo Oil is a product that made with naturally ingredients. 'Amla' juice with honey is helpful for the treatment of leucoderma. It is real that vitiligo is an terminal illness, but this does not mean you should stop living. <br><br>It is thus necessary to find a cure that can mitigate the condition and provide relief. Be careful though, sometimes lemons can irritate your skin. Acknowledged side-effects of the remedy consist of severe sensitivity to daylight, inflammation and itchy, dry pores and skin. Excessive mental stress and worrying, burn injuries, defective perspiration mechanism, ailments like typhoid fever that affect the gastrointestinal tract, acute or chronic gastric disorder, worms or parasites in the alimentary canal and impaired hepatic function are among the major contributors to this skin disorder. Here are some examples of how text messages could be employed in various industries:. |
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| He discovered algebraic identities like a<sup>3</sup>=a(a+b)(a-b) +b<sup>2</sup>(a-b) + b<sup>2</sup>.{{sfn|Tabak|2009|p=42}} He also found out the formula for <sup>n</sup>C<sub>r</sub> as [n(n-1)(n-2)...(n-r+1)]/r(r-1)(r-2)...2*1.{{sfn|Tabak|2009|p=43}} He devised formula which approximated area and perimeters of ellipses and found methods to calculate the square of a number and cube roots of a number.{{sfn|Krebs|2004|p=132}} He asserted that the [[square root]] of a [[negative number]] did not exist.{{sfn|Selin|2008|p=1268}}
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| == Rules for decomposing fractions ==
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| Mahāvīra's ''Gaṇita-sāra-saṅgraha'' gave systematic rules for expressing a fraction as the [[Egyptian fraction|sum of unit fractions]].<ref name=k497>{{Harvnb|Kusuba|2004|pp=497–516}}</ref> This follows the use of unit fractions in [[Indian mathematics]] in the Vedic period, and the [[Shulba Sutras|Śulba Sūtras]]' giving an approximation of √2 equivalent to <math>1 + \tfrac13 + \tfrac1{3\cdot4} - \tfrac1{3\cdot4\cdot34}</math>.<ref name=k497/>
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| In the ''Gaṇita-sāra-saṅgraha'' (GSS), the second section of the chapter on arithmetic is named ''kalā-savarṇa-vyavahāra'' (lit. "the operation of the reduction of fractions"). In this, the ''bhāgajāti'' section (verses 55–98) gives rules for the following:<ref name=k497/>
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| * To express 1 as the sum of ''n'' unit fractions (GSS ''kalāsavarṇa'' 75, examples in 76):<ref name=k497/>
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| {{quote|rūpāṃśakarāśīnāṃ rūpādyās triguṇitā harāḥ kramaśaḥ /<br/>
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| dvidvitryaṃśābhyastāv ādimacaramau phale rūpe //}}
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| {{quote|When the result is one, the denominators of the quantities having one as numerators are [the numbers] beginning with one and multiplied by three, in order. The first and the last are multiplied by two and two-thirds [respectively].}}
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| :: <math> 1 = \frac1{1 \cdot 2} + \frac1{3} + \frac1{3^2} + \dots + \frac1{3^{n-2}} + \frac1{\frac23 \cdot 3^{n-1}} </math>
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| * To express 1 as the sum of an odd number of unit fractions (GSS ''kalāsavarṇa'' 77):<ref name=k497/>
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| :: <math>1 = \frac1{2\cdot 3 \cdot 1/2} + \frac1{3 \cdot 4 \cdot 1/2} + \dots + \frac1{(2n-1) \cdot 2n \cdot 1/2} + \frac1{2n \cdot 1/2} </math>
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| * To express a unit fraction <math>1/q</math> as the sum of ''n'' other fractions with given numerators <math>a_1, a_2, \dots, a_n</math> (GSS ''kalāsavarṇa'' 78, examples in 79):
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| :: <math>\frac1q = \frac{a_1}{q(q+a_1)} + \frac{a_2}{(q+a_1)(q+a_1+a_2)} + \dots + \frac{a_{n-1}}{q+a_1+\dots+a_{n-2})(q+a_1+\dots+a_{n-1})} + \frac{a_n}{a_n(q+a_1+\dots+a_{n-1})}</math>
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| * To express any fraction <math>p/q</math> as a sum of unit fractions (GSS ''kalāsavarṇa'' 80, examples in 81):<ref name=k497/>
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| : Choose an integer ''i'' such that <math>\tfrac{q+i}{p}</math> is an integer ''r'', then write
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| :: <math> \frac{p}{q} = \frac{1}{r} + \frac{i}{r \cdot q} </math>
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| : and repeat the process for the second term, recursively. (Note that if ''i'' is always chosen to be the ''smallest'' such integer, this is identical to the [[greedy algorithm for Egyptian fractions]].)
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| * To express a unit fraction as the sum of two other unit fractions (GSS ''kalāsavarṇa'' 85, example in 86):<ref name=k497/>
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| :: <math>\frac1{n} = \frac1{p\cdot n} + \frac1{\frac{p\cdot n}{n-1}}</math> where <math>p</math> is to be chosen such that <math>\frac{p\cdot n}{n-1}</math> is an integer (for which <math>p</math> must be a multiple of <math>n-1</math>).
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| :: <math>\frac1{a\cdot b} = \frac1{a(a+b)} + \frac1{b(a+b)}</math>
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| * To express a fraction <math>p/q</math> as the sum of two other fractions with given numerators <math>a</math> and <math>b</math> (GSS ''kalāsavarṇa'' 87, example in 88):<ref name=k497/>
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| :: <math>\frac{p}{q} = \frac{a}{\frac{ai+b}{p}\cdot\frac{q}{i}} + \frac{b}{\frac{ai+b}{p} \cdot \frac{q}{i} \cdot{i}}</math> where <math>i</math> is to be chosen such that <math>p</math> divides <math>ai + b</math>
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| Some further rules were given in the ''Gaṇita-kaumudi'' of [[Narayana Pandit|Nārāyaṇa]] in the 14th century.<ref name=k497/>
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *Bibhutibhusan Datta and Avadhesh Narayan Singh (1962). ''History of Hindu mathematics: a source book''.
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| *{{DSB|first=David|last=Pingree|authorlink=David Pingree|title=Mahāvīra|url=http://www.encyclopedia.com/doc/1G2-2830902766.html|date=1970|ref=harv}}
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| *{{citation|last=Selin|first=Helaine|title=Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures|url=http://books.google.com/books?id=kt9DIY1g9HYC&pg=PA1268|year=2008|publisher=Springer|isbn=978-1-4020-4559-2}}
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| *{{citation|first=Takao|last=Hayashi||authorlink=Takao Hayashi|title=Mahavira|encyclopedia=Encyclopædia Britannica|url=http://www.britannica.com/EBchecked/topic/853508/Mahavira|year=2013}}
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| *{{MacTutor Biography|id=Mahavira|title=Mahavira|date=2000}}
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| *{{citation|last=Tabak|first=John|title=Algebra: Sets, Symbols, and the Language of Thought|url=http://books.google.com/books?id=h-zRieb7VbwC&pg=PA42|year=2009|publisher=Infobase Publishing|isbn=978-0-8160-6875-3}}
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| *{{citation|last=Krebs|first=Robert E.|title=Groundbreaking Scientific Experiments, Inventions, and Discoveries of the Middle Ages and the Renaissance|url=http://books.google.combooks?id=MTXdplfiz-cC&pg=PA131|year=2004|publisher=Greenwood Publishing Group|isbn=978-0-313-32433-8}}
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| *{{citation|last=Puttaswamy|first=T.K|title=Mathematical Achievements of Pre-modern Indian Mathematicians|url=http://books.google.com/books?id=DAPLaxw-53IC&pg=PA231|year=2012|publisher=Newnes|isbn=978-0-12-397938-4}}
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| * {{citation | last=Kusuba|first=Takanori | contribution=Indian Rules for the Decomposition of Fractions | year=2004 | title=Studies in the History of the Exact Sciences in Honour of David Pingree | publisher=[[Brill Publishers|Brill]] | isbn=9004132023 | issn=01698729 | editor1=Charles Burnett | editor2=Jan P. Hogendijk | editor3=Kim Plofker | editor4=Michio Yano}}
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| <!-- The article by David Pingree in the DSB has a more complete list of sources -->
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| {{Indian mathematics}}
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| {{DEFAULTSORT:Mahavira}}
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| [[Category:Indian mathematics]]
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| [[Category:Medieval Indian mathematicians]]
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| [[Category:9th-century mathematicians]]
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| [[Category:Indian Jains]]
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| [[Category:9th-century Indian people]]
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Just about 1 out of every 100 people suffers from this skin condition. People lives in country like USA, Australia and Canada known as the world. The abundance of genetic diseases associated with vitiligo clearly proves that there are a number of genes which normally govern the development and wellbeing of the melanocytes. It works by decreasing the excretion of melanin from the melanin producing cells in your skin. Even some renowned skin care and cosmetic products manufactures also claim to contain herbal oil and natural substances in their remedy.
In this connection, we will only discuss about the skin disorder when the immune system becomes dysfunctional. The signs of the white patches consist of depression or mood swings and of course the depigmentation of the vitiligo. Avoid cosmetics like cream, powder and other triggering agents. Vitiligo cure can be seen in many vitiligo skin patients who use the vitiligo natural treatment. Several herbal remedies employ various herbal components which contain psoralens, a group of naturally occurring substances.
Nevertheless, it is important to begin treatment early because older patches are harder to repigment. In US alone there are about 3 to 5 million vitiligo patients are suffering from this skin disorder disease. Infections: Infections may stimulate the onset of psoriasis. He also emphasized that he was more than proud to be black and that all the rumors that people spread were so untrue and did more than just hurt him. Re-pigmentation with Stop vitiligo Oil is very fast, good results are mostly noticeable after 2 to 3 weeks (May bee after some weeks) of common program.
But for nearly everyone people who have Vitiligo, depigmentation happens on different organs of the body (generalized pattern). Psoralea corylifolia (Babchi) is the main ingredient in the pigmento. Yellow spots, liver spots; urticaria worse from warmth of bed. He invented the Young Volumizer which is a safe procedure that is less invasive in reversing aging signs to reveal a natural and younger looking face. Your dysfunction might be owing to various tense results in.
Both conditions have a higher risk percentage in African Americans and both are immunological conditions. In most of the cases, a blend of transmissible, immunologic and ecological factors can trigger Vitiligo. Anti Vitiligo Oil is a product that made with naturally ingredients. 'Amla' juice with honey is helpful for the treatment of leucoderma. It is real that vitiligo is an terminal illness, but this does not mean you should stop living.
It is thus necessary to find a cure that can mitigate the condition and provide relief. Be careful though, sometimes lemons can irritate your skin. Acknowledged side-effects of the remedy consist of severe sensitivity to daylight, inflammation and itchy, dry pores and skin. Excessive mental stress and worrying, burn injuries, defective perspiration mechanism, ailments like typhoid fever that affect the gastrointestinal tract, acute or chronic gastric disorder, worms or parasites in the alimentary canal and impaired hepatic function are among the major contributors to this skin disorder. Here are some examples of how text messages could be employed in various industries:.